3.8 Exercise 1b

← 3.8 Exercise 1a

Exercise 1b

$For: \dot{x}=r-\text{ cosh }\left(x\right)$

  • i) Sketch the different vector field types that appear when you vary $r$. We’re actually going to do this first the hard way, then the easier way.

Looking at the phase portrait for different values of $r$ we get the following.

Figure 1

Now taking the arrows and fixed points alone and plotting them as a vector field:

Figure 2

We get the bifurcation diagram by rotating this whole plot above:

Figure 3

We can actually find these fixed points and thus the bifurcation diagram more simply. We need to solve $\dot{x}=r-\text{ cosh }\left(x\right)=0. $So the fixed points occur at:

We can plot these fixed points and this gives us

Figure 4

We actually already have whether the fixed points are stable or unstable from the vector field plot above, so let’s put these on:

Figure 5

The critical point is when the two solutions are the same (when we go from no fixed points, to one, to two):

Critical point at $r = 1$, and this occurs at: $x=0$:

Figure 6

To get the equation about the critical point into normal form we can simply expand the right hand side of the equation about $x=0.$

Defining $R=2\left(r-1$), multiplying everything by 2 and redefining $t=2T$, we get: $\frac{\text{ dx }}{\text{ dT }}=R-x^{2}$. Which is one of the two normal forms for a saddle-node bifurcation.

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